# Axiom:Axiom of Pairing/Set Theory/Strong Form

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## Axiom

For any two sets, there exists a set to which only those two sets are elements:

- $\forall a: \forall b: \exists c: \forall z: \paren {z = a \lor z = b \iff z \in c}$

That is, let $a$ and $b$ be sets.

Then there exists a set $c$ such that $c = \set {a, b}$.

Thus it is possible to create a set whose elements are two sets that have already been created.

## Also known as

The **axiom of pairing** is also known as the **axiom of the unordered pair**.

Some sources call it the **pairing axiom**.

## Also see

## Sources

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets - 1982: Alan G. Hamilton:
*Numbers, Sets and Axioms*... (previous) ... (next): $\S 4$: Set Theory: $4.2$ The Zermelo-Fraenkel axioms: $\text {ZF3}$

- Weisstein, Eric W. "Zermelo-Fraenkel Axioms." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html - Weisstein, Eric W. "Axiom of the Unordered Pair." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/AxiomoftheUnorderedPair.html